Question

Two dice are thrown. The events $A,B$ and $C$ are as follows :
$A:$ 'getting an even number on the first dice'
$B:$ 'getting an odd number on the first dice'
$C:$ 'getting the sum of numbers on the dice $\le 5$'
If $A\cap B^{\prime }\cap C$ has $n$ number of elements, then the value of $n$ is

Answer 1

$A:$ getting an even number on first dice.
$A=\left\{\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}$

$B:$ getting an odd number on first dice.
$B:\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\}$
As $B^{\prime }=A\cdots \left(i\right)$

$C:$ 'getting the sum of numbers on the dice $\le 5$'
$C=\left\{\right(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(3,1),(3,2),(2,3),(4,1)\}$

$A\cap B^{\prime }\cap C=A\cap A\cap C=A\cap C$
$\left(\text{using}\right(i\left)\right)$
$\therefore A\cap B^{\prime }\cap C=\left\{\right(2,1),(2,2),(2,3),(4,1\left)\right\}$
$\therefore n=4$